Sounds like they're fine with "turning her over" to you. *shrug* Assuming I had any sort of relationship with these folks, I would've simply told them that since they're busy with the 3 dogs they already have, that I'd be happy to take her, and hopefully that'd be that.
Legally speaking since you have all the vet records and paperwork under your name, a judge would most likely find in your favor, given that the previous owner wouldn't have a single scrap of anything with the dog's name on it.
Legally speaking since you have all the vet records and paperwork under your name, a judge would most likely find in your favor, given that the previous owner wouldn't have a single scrap of anything with the dog's name on it.